(1) If the coefficient of friction between a box and the bed of a truck is m , What is the maximum acceleration with which the truck can climb a hill, making an angle q with the horizontal, without the box's slipping on the truck bed?
(2) During the takeoff of an airplane, a passenger removes his tie and lets it hang loosely from his fingers. He observes that during the takeoff run, which last 30 seconds, the tie makes an angle of 150 with the vertical. Assume the runway is level. What is the speed of the plane at takeoff and how long the runway is needed?
(3) A particle of mass m moves in the field of a repulsive central force F (r) = Am / r3 , where A is a constant. At a very large distance from the force center the particle has speed v0 and its impact parameter is b. Show that the closest m comes to the center of force is given by rmin = (b + A / v0 )
(4) (a) Prove that the time average of the potential energy of a planet in an elliptical orbit having a semi-major axis a about the sun is -k/a, while the potential energy at a distance r is V(r)= -k/r.
(b) Calculate the time average of the kinetic energy of the planet
(5) Show that the period of a physical pendulum is equal to T = 2 √d / g where d is the distance between the point of suspension O and the center of oscillation O'.
(6) A billiard ball of radius a is initially spinning about a horizontal axis with angular speed w0 with zero forward speed. If the coefficient of sliding friction between the ball and the billiard table is before sliding ceases to occur.